i an empirical test of l r theories of price valuation e
TRANSCRIPT
SFB 649 Discussion Paper 2005-057
An empirical test of theories of price valuation
using a semiparametric approach, reference
prices, and accounting for heterogeneity
Yasemin Boztuğ*
Lutz Hildebrandt**
* Institute of Marketing, Humboldt-Universität zu Berlin, Germany ** Institute of Marketing, Humboldt-Universität zu Berlin,
Germany
This research was supported by the Deutsche Forschungsgemeinschaft through the SFB 649 "Economic Risk".
http://sfb649.wiwi.hu-berlin.de
ISSN 1860-5664
SFB 649, Humboldt-Universität zu Berlin Spandauer Straße 1, D-10178 Berlin
SFB
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Yasemin Boztuğ1 and Lutz Hildebrandt2
An empirical test of theories of price valuation
using a semiparametric approach, reference prices,
and accounting for heterogeneity
JEL Classification: C14, C23, C25, D12, M31
1 Assistant professor, Humboldt University of Berlin, Institute of Marketing, Spandauer Str. 1, 10178 Berlin, Germany, Phone: ++49-30-2093-5707, Fax: ++49-30-2093-5675, [email protected] 2 Professor, Humboldt University of Berlin, Institute of Marketing, Spandauer Str. 1, 10178 Berlin, Germany, Phone: ++49-30-2093-5691, Fax: ++49-30-2093-5675, [email protected] Financial support by the German Research Foundation (DFG) through the Sonderforschungsbereich 649, and for Boztug by way of research project #B01952/1 is gratefully acknowledged.
An empirical test of theories of price valuation
using a semiparametric approach, reference prices,
and accounting for heterogeneity
Abstract
In this paper we estimate and empirically test different behavioral theories of consumer
reference price formation. Two major theories are proposed to model the reference price reaction:
assimilation contrast theory and prospect theory. We assume that different consumer segments
will use different reference prices. The study builds on earlier research by Kalyanaram and Little
(1994); however, in contrast to their work, we use parametric and semiparametric approaches to
detect the structure of the underlying data sets. The different models are tested using a program
module in GAUSS that was able to account for heterogeneity. The model types were calibrated
by a simulation study. The calibrated modules were then used to analyze real market data.
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Several studies have shown that ignoring reference prices leads to incorrect conclusions
about consumer behavior in buying decision situations (e.g., Kalyanaram and Winer, 1995).
There is a wide body of literature dealing with different aspects of reference price formation and
its impact on price evaluation. A second stream of research uses a model-oriented approach but
builds on real buying behavior data: reference prices are specified and estimated based on
scanner panel data. In this paper, we will follow this second approach. Modeling reference price
reaction with scanner panel data leaves several issues unsolved. First, there is no general
consensus on how to derive reference prices from behavioral data (e.g., scanner panel data) (see
Winer (1988)). Second, there is no accepted dominant theory about how (or even if) the
consumer uses a reference price. Third, the true function of reference price reaction is unknown.
With respect to the latter problem, Kalyanaram and Winer (1995) called for a comparison
between different theories (especially prospect theory vs. assimilation contrast theory) to find an
acceptable solution but, until now, this call has been answered in only limited fashion. Our
research will contribute by applying a non- or semiparametric approach. Instead of testing a
certain prespecified response function with the data, our modeling approach of the consumer
choice process is driven by the structure of the data. This procedure contrasts with the common
method of assuming a theory, which is then modeled, and finally tested. However, it is well
known that interpreting parameters of misspecified models leads to wrong results and thus to
wrong recommendations for managers, a situation that can arise all too easily when using natural
data sets because the researcher does not know the correct form a priori. By contrast, in our
approach, we first detect the model structure regarding different complementary theories by using
a semiparametric method. Different specifications of reference price conceptions based on buying
data are evaluated. We also focus on a valid determination of the price effect in the underlying
choice model because, in most studies, price has the largest effect on brand choice. We also
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consider different types of consumer heterogeneity. Omitting heterogeneity may lead to the
estimation of nonvalid parameters. To incorporate all conditions into our approach, we use
nonparametric procedures in combination with common parametric methods.
We believe our research will be of both theoretical and practical relevance. Theories, even
those based on real data, are worse than useless, at least in our view, if researchers cannot or will
not make any claims about the theory’s correctness, a problem that is even more critical if the
theory is to be used for a practical purpose, such as making recommendations for real-world
decisions. This is a pitfall we are determined to avoid.
The paper is organized as follows. In the next section, we first present the theoretical
background of the reference price concept and its measurement. Next, we describe the
methodological approach of using nonparametric and parametric models. The empirical section
presents a simulation based on our methodology. We continue our empirical work with an
estimation based on real data and show that different types of consumers (loyal or nonloyal) use
different reference price systems. We end with a discussion of our results.
THE PROBLEM
In research on reference price effects, prospect theory is the most widely used approach to
explain consumer behavior. During the more than 10 years this theory has been around, it has
become the dominate approach for modeling data-based consumer reactions to price changes.
However, the question of whether prospect theory truly is the appropriate approach for modeling
consumer behavior has been insufficiently debated even though it is well known that a
misspecified model will lead to wrong parameter estimates. In most publications based on
prospect theory, validation of the theoretical concept is not mentioned.
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To obtain valid results, several modeling problems regarding reference price theory need
to be solved appropriately. First, the functional form of the price response as a part of the utility
function must be specified. For example, prospect theory assumes a linear form divided in two
parts with a kink at zero. A second problem involves exact specification of this kink, which is not
necessarily a point but could also be defined as a zone of indifference. A third important problem
concerns whether all consumers behave the same or whether there may be different price
responses by different consumers. And last, but not least, how should the reference price itself be
modeled? Are consumers able to recall prices from previous purchases or are they mainly
influenced by the price at the point of sale?
Prospect theory has appropriate solutions to some of these questions. For example, it is
possible to incorporate heterogeneity by estimating different price response parameters. Also, the
theory can integrate internal or external approaches to describing the empirical specification of a
reference price. Finally, prospect theory gives clear instructions on how to define the price
response curve, which is separated in a two-sided graph with a loss part and a gain part. This
distinction in a kinked form at zero leads to a point of indifference.
It is exactly at this point that the limitations of prospect theory become obvious. The
researcher is forced to use a two-tailed curve; no other functional form will work. Additionally,
the theory does not permit a zone of indifference, so the researcher cannot allow the consumer to
be unsure about reference points or to have no reaction to small price changes.
Our approach is based on the idea that it may be possible to choose between two theories,
namely, prospect theory and assimilation contrast theory, when describing consumer behavior. In
contrast to prospect theory, assimilation contrast theory allows for a three-tailed curve, which
includes a zone of indifference. However, similar to prospect theory, heterogeneity can be
included in a model of assimilation contrast theory, as can different reference price formulations.
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Interestingly, nearly all experimental-based approaches in the field of reference price theory use
the assimilation contrast theory. The main problem of using this theory is determining the width
of the zone of indifference sufficient to estimate a proper model. We recommend the following
two-stage procedure. First, estimate a reference price model with a nonparametric price response
function without any reference to prospect or assimilation contrast theory a priori. At the same
time, determine the width of the zone of indifference based on a free estimation of the functional
form of the price response. Second, reestimate a parametric approach with the results regarding
the correct theory of the underlying data, and, if necessary, use the nonparametrically determined
width of the indifference zone for the model. This approach leads to a data-based model while
simultaneously testing empirically for the proper theory to describe consumer behavior.
THEORETICAL BACKGROUND
The reference price concept
The reference price concept builds on the relative perception of price changes. It has been
shown that consumers in general do not only use the absolute value of price in their decisions but
also use a reference point for the valuation of different prices in their purchase decisions (e.g.,
Biswas et al. 1993; Gijsbrechts 1993; Kalyanaram and Winer 1995). This difference between the
absolute price and the reference point, or reference price, is used as an additional variable in
choice models to create a more realistic description of consumer behavior.
There are several theories concerned with reference price structure (Winer 1988). The two
most-used approaches are prospect theory (Kahneman and Tversky 1979) and assimilation
contrast theory (Sherif and Hovland 1961). The two theories differ in their assumptions about the
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influence of the reference price. Much research has been based on prospect theory even though it
is known from consumer price behavior that the reference price appears to be a region and not a
point (e.g., Emery 1969; Monroe 1971; Sawyer and Dickson 1984; Klein and Oglethorpe 1987;
Kalwani and Yim 1992). Additionally, a growing body of empirical study confirms the existence
of a latitude of acceptance, often called a price threshold (e.g., Kalyanaram and Little 1994; Han
et al. 2001; Raman and Bass 2002; Pauwels et al. 2004).
Assimilation contrast theory. Sherif and Hovland (1961) introduced assimilation contrast
theory, in which it is assumed that there is a range of indifference around the reference point. A
stimulus inside this range is perceived as smaller than its real value, an effect also known as the
assimilation effect. Assimilation contrast theory was applied to price perception by Sherif (1963).
The function of price has a cubic form, which means that a price inside the range is tolerated
(Wricke et al. 2000). A stimulus outside the range is perceived as larger than its real value, a
phenomenon known as the contrast effect. There are different assumptions about the influence of
prices outside the range. Some researchers assume that these stimuli do not have any effect on the
reference point; others think that a price outside the range, even though rejected, does have an
influence on the reference price. For a detailed discussion, see Urbany et al. (1988). Lichtenstein
and Bearden (1989) look at the range of acceptance with regard to the expected “common”
market price. They explicitly assume that the range of indifference around the reference price will
not be symmetrical.
In contrast, Kalyanaram and Little (1994) built a model based on assimilation contrast
theory integrated in a consumer choice model and assumed a symmetric range of indifference
around the reference price. They also included price variability when determining the width of
the price range. This brings us to one of the greatest difficulties in using assimilation contrast
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theory to build a choice model, which is that, usually, the width δ of the range of indifference is
unknown and must be estimated from the data or determined on the basis of experiments or
questionnaires. We will address this problem in more detail later in the paper.
Prospect theory. In contrast to assimilation contrast theory, prospect theory (Kahneman
and Tversky 1979) assumes a range of zero around the reference price; the “zone” thus becomes
a point. Prospect theory is used to describe decisions under risk (Edwards 1996). The price
decision can be seen as a risky decision (Urbany and Dickson 1990) and so the concepts of
prospect theory can be used to evaluate prices. With Thaler’s (1985) introduction of the theory of
mental accounting, it became possible to generate different price scenarios as well.
The key points of prospect theory have to do with the different valuation of gains and
losses, which are defined, respectively, as a positive or negative value of the reference point. It is
assumed that the value function describing the difference is concave for gains and convex for
losses. Also, the function is steeper for losses than for gains. This behavior is called loss aversion.
For a more detailed description of prospect theory, the reader is referred to Edwards (1996),
Kahneman and Tversky (1979), Thaler (1980), or Tversky and Kahneman (1981). Using the
concept of mental accounting, the value function of prospect theory is expanded. The total utility
of a purchase is divided between an acquisition utility and a transaction utility. The latter can be
described as the difference between the reference price and the paid price.
Empirical modeling. The reference price is a latent construct that cannot be measured
directly. It can be determined only by experiment or by asking consumers about their expected
price (for an overview, see, e.g., Estelami et al. 2001), procedures that have their own problems.
Scanner data do not provide this sort of information; however, such data do provide a different
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type of information that can be used to estimate a reference price. There are two ways of
accomplishing this. One method uses prices paid in the past and the other uses current prices. In
the first approach, it is assumed that consumers use prices paid in the past to build their reference
price; this is called an “internal reference price” (e.g., Winer 1986). In the second approach, it is
assumed that consumers use prices at the point of sale, which are actual prices; this is called an
“external reference price” (e.g., Hardie et al. 1993). For an empirical comparison of both
methods, see, for example, Briesch et al. (1997b).
Obviously, only consumers able to recall past prices can use the internal reference price.
These people are considered to be highly involved shoppers or price-sensitive consumers. These
consumers have an internal reference price for each separate brand of merchandise. There are
various formulations for the quantification of an internal reference price, but none of them have
become standard (e.g., Wricke et al. 2000).
The external reference price is based on current prices at the point of purchase. It is not
brand specific, but only category specific. It is assumed that the consumer does not use any
information on past purchases, perhaps because he or she cannot remember past prices paid or
does not trust his or her memory of them. Therefore, this type of consumer uses only information
that has become available during the current shopping trip. As with the internal reference price,
the literature provides no common calculation method for an external reference price.
Research question
Table 1 summarizes all possible model type and theory combinations. We sorted the
models by their underlying behavioral theories, the reference price concept utilized, and by type
of model specification. The first model based on a reference price concept was introduced by
8
Winer (1986), who used only the internal reference price construct. We mention only a few of the
many studies based on prospect theory. This model type rarely uses controls, seemingly assuming
that the theoretical construct is in complete agreement with consumers’ real behavior. We will
take a critical look at this assumption later in our paper.
—————————————————
Table 1 about here.
—————————————————
Assimilation contrast theory within a parametric estimation is presented by Kalyanaram
and Little (1994). They focus on internal reference price constructs. A limited comparison is
made between prospect theory and assimilation contrast theory.
Models based on a semiparametric approach are estimated by Abe (1998). In estimating
the width of the indifference zone, he uses a model similar to assimilation contrast theory. His
approach can be regarded as a semiparametric extension of Kalyanaram and Little (1994).
However, Abe (1998) uses only an internal reference price specification and does not account for
heterogeneity. Also, there is no comparison made between prospect and assimilation contrast
theory. A recent approach of Pauwels et al. (2004) uses internal and external reference prices,
using only a parametric approach, and, again, there is no comparison made between the two
theories.
As Table 1 clearly shows, numerous research questions (marked by an × ) are still
unresolved. Our study will fill in some of these gaps. We decide data-driven which theory is the
appropriate one. We investigate which model is best when it is matched (or even not matched)
with simulated and real data. Our aim is to be able to tell which theory will be the appropriate one
9
in specific situations or for specific consumers. From the methodological point of view, we build
up a semiparametric model that is able to create a model based on either prospect theory or on
assimilation contrast theory. We use different internal and external reference price specifications,
making our model much broader than those used in most reference price studies. Our most
important contribution is the comparison of prospect theory with assimilation contrast theory
using both parametric and semiparametric approachs.
THE METHODOLOGICAL APPROACH
Based on disaggregated data, we use a consumer-based approach. The general model
structure can be described as
systematic utility random componentin in inU V ε= + .
Different assumptions can be specified for the systematic and random components. Both
components can be modeled in a parametric or a nonparametric way, resulting in four model
types. We will focus on only two models. In one of these, both components are described in a
parametric way and the model can be formulated with a multinomial logit model. The other
model we focus on is a semiparametric one resulting from a nonparametric formulation of the
systematic part and a parametric modeling of the random component. A suitable model
description for this type is a generalized additive model.
Parametric approach
A well-known disaggregated choice model is the multinomial logit model (MNL), which
was introduced in the marketing context by Guadagni and Little (1983). Here, both the systematic
component and the random component are modeled in a parametric way. For the difference of the
10
errors, a Gumbel distribution is assumed. The model structure is linear-additive. A common
MNL is:
(1) exp( )( )Pr exp( )n
inn
jnj C
xix
ββ
∈
′=
′∑,
with Prn(i) the probability of individual n to buy product i, xin the explanatory variables of
product i for consumer n, and ß the parameter to be estimated. The approach is estimated with
maximum-likelihood.
Semiparametric approach
An alternative approach to the MNL model could be a non- or semiparametric method.
Here, we focus on a generalized additive model (GAM) (Hastie and Tibshirani 1990). This
method has several advantages. The error structure can be modeled as in the MNL model, thus
making possible the comparison of the estimation results. The systematic component of a GAM
has, as does the MNL, an additive structure. The main difference from a MNL is a one-
dimensional nonparametric function for each explanatory variable, in contrast to a linear
modeling of the explanatory variables in a MNL. The formal model is presented in Equation (2).
(2) 1E[ ] ( ) with1 exp( )p p
pY X x G f x G
x
| = = = + − ∑
Here, fp are one-dimensional nonparametric functions to be estimated and xp are the
explanatory variables; G is called a link function. By specifying G as in Equation (2), the error
term distribution of the GAM is equal to a common MNL approach. The standard GAM is
defined only for continuous explanatory variables. In marketing applications, there are usually
also binary explanatory variables, for example, display or feature. To incorporate these variables,
the common GAM is extended to include a linear-additive component, as in the MNL model, at
11
least for all binary variables. This model type is described in Equation (3).
(3) E[ ] ( )p pp
Y X x G f x xβ
′| = = + ∑
Two different algorithms exist for estimating a GAM. The most common approach is
backfitting, introduced by Friedman and Stuetzle (1981). The method is based on a variance
decomposition of the total variance to all explanatory variables specified in the model. An
alternative method is called marginal integration, where the marginal influences of the
explanatory variables to the response variable are estimated. Due to the empirical focus of our
research, we do not describe the mathematical structure of the estimation procedures in detail but
instead refer the reader to Hastie and Tibshirani (1990) or Linton and Nielsen (1995).
Integration of the reference price concept in a parametric model
Integrating prospect theory into a common choice model means adding two components,
which, respectively describe the loss or gain perceived by the consumer. The pure price
component can be included only in a model with an internal reference price formulation. In
modeling an external reference price approach, the price component has to be excluded due to
strong correlation with the gain and/or loss components (e.g., Hardie et al. 1993). The resulting
model describing the utility Uint of consumer i for product n at time t is shown in Equation (4).
(4)
0 1 2
3
4 5 6
( )
( )
int int
int int
int int int int P IRP
Loss
int int IRP P
Gain
int int intint
U P P IRP I
IRP P I
DisplLOY Feat
β β β
β
β β β
>
>
= + + −
+ −
+ + + ε+
The gain and loss components come into play only when the internal reference price is
larger or smaller than the actual price. All other explanatory variables are included as in a
standard logit model, for example, P for price, LOY for loyalty, as specified by Guadagni and
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Little (1983), Displ for the binary variable for the existence of display, and, in the same manner,
Feat for feature. ß describes all parameters to be estimated including the brand dummies ß0.
In an assimilation contrast model, a mid range, also known as a range of indifference,
must be included. Here, we assume that for the contrast effects, we again have loss and gain
components (Kalyanaram and Little 1994).3 We make this assumption for two reasons. First, the
model based on assimilation contrast theory reduces to a prospect theory model if the mid
component (range of indifference) is zero, thus making it easy to compare of two models.
Second, even if a range of indifference is assumed, it can be assumed, additionally, that the
consumers will display loss aversion.
In the formal representation of the assimilation contrast model (Equation (5)), in addition
to the common variables, a parameter, δ , is specified. δ describes the width of the range of
indifference. In the model presented here, we assume a symmetric indifference zone around the
price gap (loss minus gain) (e.g., Kalyanaram and Little 1994).
(5)
0 1 2
3 2
4 2
5 6
( )
( )
( )
int int
int int int
int int
int int int int P IRP
Loss
int int IRP P IRP
Mid
int int P IRP
Gain
int int
U P P IRP I
P IRP I
IRP P I
LOY
2
2
δ
δ δ
δ
β β β
β
β
β β
> + /
− / < < + /
< − /
= + + −
+ −
+ −
+ + 7 int intDispl Featβ ε+ +
In a parametric approach, it is nearly impossible to detect δ if it is assumed to be
nonsymmetrical. Even in the symmetric case, a grid search for δ is extensive. One possible way
3 Han et al. (2001), Raman and Bass (2002), and Pauwels et al. (2003) used models with price threshold components. The price threshold models are not clearly based on assimilation contrast theory; they seem to be more data driven (Raman and Bass present some theory, but their model is on an aggregate level and they use only a reference price
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to find a nonsymmetrical δ is to use a nonparametric approach.
inU
Integration of the reference price concept in a semiparametric approach
The semiparametric estimation combines nonparametric and parametric terms in the
specification of the utility function. All continuous explanatory variables can be modeled with a
one-dimensional nonparametric function. In a reference price context, these variables are loyalty,
price gap, and price. Equation (6) does not include a nonparametric function for price because the
price component is used only in models with an internal reference price effect (see the section on
prospect theory above).
(6) 0 1 1 2
2 3
( ) ( intt int int
intint
P f pricegap f LOYFeatDispl
β ββ β ε
= + + +
+ + +
)
In many models, using a nonparametric function for loyalty leads to a quasi-linear
functional form. The main nonparametric component is the price gap. Estimating the functional
form of the price gap leads to a very flexible form. Therefore, no explicit specification of gain,
loss, or an indifference zone need be made. Additionally, it is not necessary to estimate a
threshold parameter. By inspecting the estimated functional form of the price gap, the researcher
will be able to determine which theory (e.g., prospect or assimilation contrast) is the appropriate
one for the estimated data set.4
THE APPLICATION
Simulation study
We estimated the semiparametric function using a program written in Gauss. To
demonstrate the validity of our estimation, a simulation study was conducted with three goals in
and not the difference between price and the reference price). 4 For example, Pauwels et al. (2003) mentioned that fully nonparametric models have high data requirements, which
14
mind. First, we wanted to demonstrate that using the same theory for simulating and estimating
the data set will result in appropriate estimates. Second, we wanted to show what would happen if
simulated data based on one theory, for example, assimilation contrast theory, is used in an
estimation made with a model using the other theory, vice versa. Third, we wanted to illustrate
that if one uses the semiparametric approach, it is not necessary to specify any particular theory;
the functional form will be enough to detect the theory underlying the data set.
For the simulation we generated data sets with two different sample sizes: 2,000 and
5,000 purchases for 200 households. Each data set contains three brands and, as explanatory
variables, price, promotional price, and feature. The study is similar in design to that of Chang et
al. (1999); however, in our study, we included internal or external reference prices and build our
model in reference to prospect theory and assimilation contrast theory. In sum, we have eight
different data sets, which are estimated based on either the underlying theory, the opposite theory,
or with the semiparametric approach, resulting in 24 different results. Variable specifications are
laid out in Table 2.
—————————————————
Table 2 about here.
—————————————————
For the data set based on assimilation contrast theory, we assumed a nonsymmetrical
range of indifference with gain 0 2δ = . and loss 0 1δ = . . This asymmetric assumption follows from
Kalyanaram and Little (1994), who proposed that a nonsymmetrical range is much more
plausible than a symmetric one. According to prospect theory, consumers will react more
strongly to a loss than to a gain, which leads to the assumption that the indifference zone should
be smaller on the loss side than it is on the gain side.
15is not true for our approach because we model only one component nonparametrically.
Model based on prospect theory. The first simulation study results are from the data sets
based on prospect theory. In this article, we document the results for 5,000 observations and the
use of an internal reference price. All other results are available on request to the first author. In
Table 3 and Figure 1 we present the parametric and nonparametric results.
—————————————————
Table 3 about here.
—————————————————
—————————————————
Figure 1 about here.
—————————————————
Table 3 illustrates that when estimating a model based on prospect theory, all true
parameter values lie within a 95% approximate confidence interval and are all significant. By
using a model based on assimilation contrast theory, the grid search for δ leads to an “optimal”
value of 0, so the model reduces to one regarding prospect theory. Of course, all assimilation
contrast theory parameter estimates are identical to the prospect theory parameters. If we use an
approach without any reference price specification, for example, a standard logit model, both
estimated parameter values are still significant, but the approximate confidence interval for the
price component no longer includes the true parameter.
For the semiparametric model, we specify price and feature in a parametric manner, and
the price gap (with loss minus gain) in a nonparametric way. This procedure, and the fact that the
estimates are in a nonclosed form, means that the estimation result for price gap can be presented
only pictorially, which is done in Figure 1. The parameter estimates are all significant and the
approximately confidence interval includes the true parameter values with 95% probability.
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Figure 1 shows that for the price gap, all estimations are close together, leading to the conclusion
that in this case, use of prospect theory is appropriate.
Model based on assimilation contrast theory. Again, we present results only for the data
set with 5,000 observations. The indifference zone for the simulated data is asymmetric. Two
different models based on assimilation contrast theory were estimated: one with the true
asymmetric values for δ and one with a symmetric δ value, as suggested by Kalyanaram and
Little (1994). All parametric estimation results are presented in Table 4.
—————————————————
Table 4 about here.
—————————————————
For both estimations based on assimilation contrast theory, we get significant parameter
estimates (except for the mid range, as expected), and all approximate confidence intervals
include the true parameter values. In the model with the true δ values, the estimated parameters
are closer to the real values than are the estimates obtained using the symmetric δ values; the
loglikelihood is also better in this model than in the one with symmetric values for δ . In the
model with a symmetric indifference zone, we find an optimal width of 0.2, which is smaller than
the original value. A look at the estimates resulting from a model based on prospect theory
reveals several interesting results. First, every parameter estimate is still significant. Second, the
approximate confidence intervals of loss and gain do not include the true parameter values,
something that would not be obvious in an application using real data. So, although the
significance estimates appear to be a good representation of the data, the estimates are in actuality
not even close to the true underlying parameters. Therefore, parametric estimation with the wrong
model may still generate significant parameters—despite misspecification.
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We now turn to the results of the semiparametric approach. Again, both parametrically
estimated parameters are significant and their approximate confidence intervals include the true
values. The nonparametric estimates are shown in Figure 2.
—————————————————
Figure 2 about here.
—————————————————
The functional form of the nonparametric estimate is close to the real values, as is the
asymmetric indifference zone; clearly, assimilation contrast theory is at work. Figure 2 illustrates
the boundaries of the indifference zone. From the figure, it is obvious that prospect theory’s
estimates are wrong, especially in the zone around the price gap, which is the most relevant area
for price discounts. Assimilation contrast theory’s estimation of a too-small symmetric
indifference zone is not perfect, either, but it is not as bad as prospect theory estimates.
Several conclusions can be drawn from the simulation study results. Estimating a data set
with the correct underlying theory results in significant parameter estimates that also include the
true parameters in an approximate confidence interval. Using the “wrong” theory can produce
three different results. First, if prospect theory is the underlying theory, even estimating a model
based on assimilation contrast theory will lead to correct results because an “optimal” width of
zero will be estimated for the indifference zone. Second, and most important, using a prospect-
theory-based model for estimation of a data set that resulted from assimilation contrast theory
leads to significant parameter estimates, but the approximate confidence interval will not include
the true parameters. This fact cannot be detected for a real data set. Third, regardless of which
theory is the “correct” one for a data set, the semiparametric estimates will give a clear indication
of which model based on which theory should be used. If it turns out that the semiparametric
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estimates recommend the use of assimilation contrast theory, nonparametric estimates could be
applied so as to provide some guidance on the width and even the symmetry of the indifference
zone.
Real data set
We next used the Gauss program to estimate the models using real data. The database
consists of purchases of one product category in a micro test market. The data cover 104 weeks
and seven brands of merchandise from a major retail store. During the specified time period, 876
households made 3,647 purchases. For calibration purposes, we split off the data for the first 80
weeks. The remaining 24 weeks are used for validation. The data consisted of purchase
information about the actual price paid and the existence of display and feature actions. From
these data, we constructed the loyalty variable proposed by Guadagni and Little (1983) and for
each reference price concept (internal and external) three different specifications, which are
briefly explained below.
The internal reference price can be based on several alternative concepts of memory. The
price of the last period (e.g., Kalwani et al. 1990) uses only the information about the last
purchase. Here, it is assumed that the consumer does not have or use any price information from
purchases made previous to the last purchase. Formally, it can be described as 1int intIRP P −= , with
i the brand, n the consumer, and t the time index. Consumers using the mean of last prices (e.g.,
Rajendran and Tellis 1994), use not only the last price paid in calculating the internal reference
price, but also all other prices paid until time period T. It is implicitly assumed that the consumer
has a long price memory. T can be set to a specific value, for example, 3 or 4, so only the last 3 or
4 time periods are included in the calculation, with 1
1 T
int intIRP PT τ
τ−
=
= ∑ . In the weighted mean of
19
last prices (e.g., Winer 1985), not all former prices have the same weight in the calculation, and
to add to the complexity, the weight they will have must be estimated. This form of the internal
reference price is similar to loyalty measure introduced by Guadagni and Little (1983) and can be
modeled as . ( )11 11
intint int intIRP IRP P IRPζ−− − = + − −
( )1 11int intLOY yλ λ− −= + −
External reference prices are based on price information at the point of purchase. They are
category specific only. The lowest price in the category (e.g., Biswas and Blair 1991) is often
used as an external reference price because consumers merely focus on lower rather upper price
limits for comparison. Formally, it can be written as { }minnt itiERP P= . The actual price of last
brand bought (e.g., Hardie et al. 1993) is used in cases when consumers cannot remember the last
price paid, but can remember the last brand bought. It can be modeled as if i was
bought in t – 1. The weighted price with loyalty (e.g., Mazumdar and Papatla 1995) is an
extension of the price of the last brand bought. Here, not only the last brand bought, but also all
purchased brands, are taken into account in calculating the external reference price. These prices
are weighted with the loyalty felt toward each purchased brand. By weighting the actual prices
with the loyalty, this type of external reference price can be modeled as with
.
nt itERP P=
1
I
nti
ERP=
= ∑ int itLOY P
intLOY
In this paper, we present only the reference price specification that leads to the best model
fit regarding the log-likelihood value. When we turn to discussing heterogeneity, we will use only
the results dealing with the loyalty variable (e.g., Abramson et al. 2000; Kamakura et al. 1996).
Table 5 is a summary of all parametric estimation results without incorporating
heterogeneity, except for the loyalty variable. Only the results for the best internal and external
20
reference price formulations are shown: the weighted past prices for the internal reference price
and the actual price of the last brand bought for the external reference price. Both reference price
formulations are used because it is not obvious which one describes consumer behavior most
appropriately.
For ease of comparison, we first look at all estimations using the internal reference price.
The parameter estimates for loyalty, display, and feature are very similar for both the model
based on prospect theory and the model based on assimilation contrast theory. Larger differences
between the models can be seen for all components related to price (price itself, gain, loss, and
mid). Most interesting is the highly significant mid region in the assimilation contrast model, for
which we found an “optimal” width of 0.2. This implies that it is not its original functional form,
which has an insignificant mid region, that is the proper form of assimilation contrast theory, but
instead a functional form with a very steep mid region and much flatter edges. This form leads to
a model based on prospect theory, but with a slightly different function course. Furthermore, the
nonparametric estimation is similar to the model based on assimilation contrast theory, as can be
seen in Figure 3.
—————————————————
Table 5 about here.
—————————————————
—————————————————
Figure 3 about here.
—————————————————
Looking now at the results for the external reference price, prospect theory appears to be
the most appropriate in both parametric models because we find an optimal width of 0 for the
21
indifference zone. From Figure 4, which also displays the nonparametric estimate, it is obvious
that all estimates lie very close together.
—————————————————
Figure 4 about here.
—————————————————
Consideration of heterogeneity. We now extend our study to estimate the effect of
heterogeneity as ignoring it could lead to wrong parameter estimates (e.g., Chang et al. 1999).
For a brief presentation of different heterogeneity concepts, the reader is referred to DeSarbo et
al. (1999). We follow the approaches of Krishnamurthi and Raj (1991), Krishnamurthi et al.
(1992), and Mazumdar and Papatla (1995). These authors make an a priori segmentation of loyal
and nonloyal consumers.5 Due to the assumption that loyal consumers are not very price
sensitive, we assume that they use an external reference price. Nonloyal consumers are much
more price-conscious and thus are able to use an internal reference price. We use the same model
type for internal and external reference price formulation as was used in the models without
heterogeneity.
In our model formulations, we focus on a segment-specific level. We account for
heterogeneity of preference by including the loyalty variable. Heterogeneity of response is
captured by brand intercepts; structural heterogeneity through the segmentation of consumers.
In Table 6, we first look at the parametrically estimated results. The model based on
5 We also used a latent class approach (Kamakura and Russell 1989; Jain et al. 1994) limited to two segments, but the results are close to the a priori segmentation. The results of the a priori segmentation can be interpreted more easily so only those results are used in the following. Another reason for not using the latent class approach is that in that method, group membership is only weakly related to secondary drivers data (DeSarbo et al. 1997). Erdem et al. (2001) explicitly demonstrated the problems that arise when using a latent class approach for a reference price model.
22
prospect theory shows different parameter values for both consumer segments, particularly for all
price components (e.g., price, loss, and gain) and for the loyalty variable. Compared to the model
without heterogeneity (see Table 5), the values for display and feature stayed approximately
constant, but loyalty can be seen as a separational index. There are also some differences between
the two models for loss and gain; the loss values are bigger, as is the gain value, for the internal
reference price model. In the model for loyal consumers (ERP), the gain component is no longer
significant.
—————————————————
Table 6 about here.
—————————————————
The model based on assimilation contrast theory is similar to the prospect theory model in
that all price component values are larger. However, for loyal consumers, gain and the mid
component are not significant. The optimal width of the indifference zone is 0.1, which is exactly
the mean of 0.2 and 0, the widths of this zone in the models without heterogeneity.
Using a nonparametric estimation for the price gap results in different loyalty values, just
as it did when the parametric approaches were employed. Figure 5 shows that there is not much
difference between this model and the one without heterogeneity (Figure 3) for nonloyal
consumers. However, the estimates for loyal consumers (Figure 6) changed dramatically. The
model illustrated in Figure 6 resembles prospect theory, but it is clear that there is an area (i.e.,
the zone of indifference) where consumers do not react to price changes.
—————————————————
Figure 5 about here.
—————————————————
23
—————————————————
Figure 6 about here.
—————————————————
As discussed previously, the nonparametric estimates are used in two ways. First, we
want to detect which theory is best represents the underlying data set. Second, if the first step
shows assimilation contrast to be the best theory, we additionally use the nonparametric estimates
to obtain an optimal δ for the width of the zone of indifference. This makes it possible to
abandon the restriction of a symmetric indifference zone. Of course, a visually observed range
will never be exact, but it can lead to much better estimation results, especially when using a
nonsymmetrical indifference zone.
From Figures 5 and 6, it can be determined that for the segment of nonloyal consumers
using an internal reference price, the zone of indifference is symmetric interval with a width of
0.14. Loyal consumers using an external reference price have a nonsymmetrical zone, with a
width of 0.1 on the gain side and 0.04 on the loss side. With these new values for δ we
reestimate our model incorporating heterogeneity by separating consumers into loyal and
nonloyal.
—————————————————
Table 7 about here.
—————————————————-
Table 7 compares the estimation based on the assimilation contrast theory model with
symmetric and, due to the parametric model restriction, “suboptimal” δ with the model using the
δ values extracted from the Figures 5 and 6. Due to a slightly larger zone of indifference in the
model with δ from the nonparametric estimation, all price-related values change slightly, the
24
largest changes being to the gain and mid components in the external reference price model. Most
important is the significant improvement in model fit, which can also be found in the prediction
samples.
DISCUSSION
When describing consumer behavior in regard to brand choice and price decisions,
reference price should be added to the usual multinomial logit model so to incorporate
nonrational behavior. Generally, two theories are used to model this behavior: prospect theory
and assimilation contrast theory. These theories differ in many ways, but here we concentrated
only on the existence or nonexistence of an indifference zone in the utility function related to
price. Most reference price studies use models based on prospect theory. In our simulation study
we showed that using prospect theory for data sets based on assimilation contrast theory leads to
significant parameter estimates, but that these estimates do not include the true values in the
approximate confidence interval. In the real data set application, we used a segmentation of two
consumer types (loyal and nonloyal). The model estimation was done not only with parametric
approaches based on either prospect theory or assimilation contrast theory, but also with a
nonparametric approach, where no theory specification is necessary. We found that neglecting
consumer heterogeneity could lead to overlapping effects regarding which theory is the “correct.”
Loyal and nonloyal consumers have different reactions to price promotions. Loyal consumers
behave according to assimilation contrast theory and exhibit a nonsymmetrical indifference zone,
the detection of which is possible under a nonparametric estimation method. Nonloyal consumers
act consistent with prospect theory. For this segment of consumers, we found a very steep part
around the zero region of the price gap and less steep areas for loss and gain.
25
Our study makes a significant contribution to the reference price literature by showing
what can happen when the “wrong” theory is used to predict consumer behavior, the implications
of which we believe have many practical applications. Management, in particular, will be keenly
interested in correctly estimating consumers’ indifference zones before making important and
potentially expensive merchandizing decisions.
26
Psychological theory Comparison
Type of model
Reference
price type
Prospect theory
Assimilation
contrast theory
Prospect versus
assimilation
contrast theory
parametric
IRP e.g., Winer (1986) e.g., Kalyanaram
and Little (1994)
Kalyanaram and
Little (1994)
ERP e.g., Hardie et al. (1993) × ×
both e.g., Briesch et al. (1997a),
Mazumdar and Papatla
(2000)
Pauwels et al.
(2004)
×
semiparametric
IRP × Abe (1998) ×
ERP × × ×
Table 1: Overview of all possible model type and theory combinations
27
Brand 1 Brand 2 Brand 3
Price uniform [0.75; 1.00] uniform [0.65; 0.90] uniform [0.60; 0.70]
Promoted price uniform [0.50; 0.70] uniform [0.40; 0.55] uniform [0.45; 0.55]
Promotional
frequency
1 out of 4 weeks 1 out of 4 weeks 1 out of 8 weeks
Feature frequency 1 out of 16 weeks 1 out of 16 weeks 1 out of 36 weeks
Table 2: Variable specification for simulation study
28
Parametric results
True
model
Prospect
theory
Assimilation
contrast
theory
Without
reference
price
Semiparametric
approach
P -7 -7.42*† -7.42*† -11.37* -7.40*†
Feat 2 2.16*† 2.16*† 1.83*† 2.14*†
Gain 4 4.33*† 4.33*† n. a.
Loss -6.5 -6.28*† -6.28*† n. a. Plot
δ 0 n. a. 0 n. a. Plot
-2263 -2263 -2538 -2262
2ρ 0.495 0.495 0.435 0.495
†True parameter value lies with 95% probability in the approximate confidence interval.
*Significant.
Table 3: Estimation results for simulated data set based on prospect theory
29
Parametric results
True
model
Assimilation
contrast
theory
Assimilation
contrast
theory
Prospect
theory
Semiparametric
approach
P -7 -7.42*† -7.85*† -7.60*† -7.26*†
Feat 2 2.02*† 1.87*† 1.83*† 2.03*†
Gain 4 3.92*† 3.21*† 2.96*
Mid 0.05 0.52 -0.68 n. a.
Loss -6.5 -6.24*† -5.98*† -5.30*
Plot
δ 0.2 / 0.1 0.2 / 0.1 0.1 / 0.1 n. a. Plot
-2383 -2409 -2434 -2386
2ρ 0.469 0.463 0.458 0.468
†True parameter value lies with 95% probability in the approximate confidence interval.
*Significant.
Table 4: Estimation results for simulated data set based on assimilation contrast theory
30
Prospect theory Assimilation contrast
theory
Semiparametric
approach
IRP ERP IRP ERP IRP ERP
Without
reference
price
P -2.93* n. a. -2.38* n. a. -1.68* n. a. -4.68*
Displ 0.64* 0.62* 0.60* 0.62* 0.58* 0.60* 0.63*
Feat 1.29* 1.29* 1.14* 1.29* 1.11* 1.19* 1.28*
Gain 1.34* 2.76* 1.68* 2.76* n. a.
Mid n. a. n. a. -9.09* n. a. n. a.
Loss -3.29* -6.05* -1.98* -6.05*
Plot
Plot
n. a.
LOY 5.09* 5.30* 5.09* 5.30* 5.28* 5.31* 5.15*
δ n. a. n. a. 0.2 0 Plot Plot n. a.
-3845 -3836 -3810 -3836 -3794 -3812 -3852
2ρ 0.270 0.272 0.277 0.272 0.279 0.275 0.269
*Significant.
Table 5: Estimation results for real data set without incorporating heterogeneity
31
Prospect theory Assimilation contrast
theory
Semiparametric
approach
IRP ERP IRP ERP IRP ERP
P -3.55* n. a. -2.04* n. a. -0.87* n. a.
Displ 0.69* 0.58* 0.67* 0.58* 0.61* 0.57*
Feat 1.27* 1.01* 1.24* 1.06* 1.10* 0.98*
Gain 3.22* 0.20 4.08* 0.05
Mid n. a. n. a. -14.39* -5.22
Loss -3.99* -7.64* -3.47* -6.75*
Plot
Plot
LOY 2.55* 7.34* 2.72* 8.41* 2.50* 7.51*
δ n. a. 0.1 Plot
-3589 -3581 -3545
2ρ 0.317 0.318 0.324
*Significant.
Table 6: Estimation results for real data set incorporating heterogeneity
32
Assimilation contrast
theory δ suboptimal
Assimilation contrast
theory δ optimal
IRP ERP IRP ERP
P -2.04* n. a. -1.80* n. a.
Displ 0.67* 0.58* 0.64* 0.60*
Feat 1.24* 1.06* 1.24* 1.03*
Gain 4.08* 0.05 4.34* 0.77
Mid -14.39* -5.22 -13.41* 2.61
Loss -3.47* -6.75* -1.94 -7.21*
LOY 2.72* 8.41* 2.84* 8.44*
δ 0.1 0.1 0.14 0.1 / 0.04
-3581 -3575
2ρ 0.318 0.319
*Significant.
Table 7: Estimation results for real data set incorporating heterogeneity based on assimilation
contrast theory, with parametrically and nonparametrically produced δ values
33
Figure 1: Semiparametric and parametric estimation for the price gap in the simulation study with
data based on prospect theory
34
Figure 2: Semiparametric and parametric estimation for the price gap in the simulation study with
data based on assimilation contrast theory
35
Figure 3: Semiparametric and parametric estimation for the price gap in the real data set for the
internal reference price without incorporating heterogeneity
36
Figure 4: Semiparametric and parametric estimation for the price gap in the real data set for the
external reference price without incorporating heterogeneity
37
Figure 5: Semiparametric and parametric estimation for the price gap in the real data set for
nonloyal consumers incorporating heterogeneity
38
Figure 6: Semiparametric and parametric estimation for the price gap in the real data set for loyal
consumers incorporating heterogeneity
39
References
Abe, Makoto (1998), “Measuring consumer, nonlinear brand choice response to price,”
Journal of Retailing, 74(4), 541–68.
Abramson, Charles, Rick L. Andrews, and Imran S. Currim (2000), “Parameter bias from
unobserved effects in the multinomial logit model of consumer choice,” Journal of Marketing
Research, 37, 410–26.
Biswas, Abhijit, and Edward A. Blair (1991), “Contextual effects of reference price in
retail advertisements,” Journal of Marketing, 55(3), 1–12.
----, Elizabeth J. Wilson, and Jane W. Licata (1993), “Reference pricing studies in
marketing: A synthesis of research results,” Journal of Business Research, 27, 239–56.
Briesch, Richard A., Pradeep K. Chintagunta, and Rosa L. Matzkin (1997a),
“Nonparametric and semiparametric models of brand choice behavior,” Discussion paper,
University of Texas at Austin.
----, Lakshman Krishnamurthi, Tridib Mazumdar, and S. P. Raj (1997b), “A comparative
analysis of reference price models,” Journal of Consumer Research, 24, 202–14.
Chang, Kwangpil, S. Siddarth, and Charles B. Weinberg (1999), “The impact of
heterogeneity in purchase timing and price responsiveness on estimating of sticker shock effects,”
Marketing Science, 18(2), 178–92.
DeSarbo, Wayne S., Asim Ansari, Pradeep Chintagunta, Charles Himmelberg, Kamel
Jedidi, Richard Johnson, Wagner Kamakura, Peter Lenk, Kannan Srinivasan, and Michel Wedel
(1997), “Representing heterogeneity in consumer response models,” Marketing Letters, 8(3),
335–48.
----, Youngchan Kim, and Duncan Fong (1999), “A Bayesian multidimensional scaling
40
procedure for the spatial analysis of revealed choice data,” Journal of Econometrics, 89, 79–108.
Edwards, Kimberly D. (1996), “Prospect theory: A literature review,” International
Review of Financial Analysis, 5(1), 19–38.
Emery, Fred (1969), “Some psychological aspects of price,” in Pricing Strategy, Bernard
Taylor and Gordon Wills, eds. London: Staples.
Erdem, Tülin, Glenn Mayhew, and Baohang Sun (2001), “Understanding reference-price
shoppers: A within- and cross-category analysis,” Journal of Marketing Research, 38, 445–57.
Estelami, Hooman, Donald R. Lehmann, and Alfred C. Holden (2001) “Macro-economic
determinants of consumer price knowledge: A meta-analysis of four decades of research,”
International Journal of Research in Marketing, 18, 341–55.
Gijsbrechts, Els (1993), “Prices and pricing research in consumer marketing: Some recent
developments,” International Journal of Research in Marketing, 10, 115–51.
Guadagni, Peter M., and John D. C. Little (1983), “A logit model of brand choice
calibrated on scanner data,” Marketing Science, 2(3), 203–38.
Han, Sangman, Sunil Gupta, and Donald R. Lehmann (2001), “Consumer price sensitivity
and price thresholds,” Journal of Retailing, 77, 435–56.
Hardie, Bruce G. S., Eric J. Johnson, and Peter S. Fader (1993), “Modeling loss aversion
and reference dependence effects on brand choice,” Marketing Science, 12(4), 378–94.
Hastie, Trevor, and Robert Tibshirani (1990), Generalized Additive Models. London:
Chapman & Hall.
Jain, Dipak C., Naufel J. Vilcassim, and Pradeep K. Chintagunta (1994), “A random-
coefficients logit brand-choice model applied to panel data,” Journal of Business & Economic
Statistics, 12(3), 317–28.
Kahneman, Daniel, and Amos Tversky (1979), “Prospect theory: An analysis of decision
41
under risk,” Econometrica, 47(2), 263–91.
Kalwani, Manohar U., Chi Kin Yim, Heikki J. Rinne, and Yoshi Sugita (1990), “A price
expectations model of customer brand choice,” Journal of Marketing Research, 17, 251–62.
----, and ---- (1992), “Consumer price and promotion expectations: An experimental
study,” Journal of Marketing Research, 29, 90–100.
Kalyanaram, Gurumurthy, and John D. C. Little (1994), “An empirical analysis of latitude
of price acceptance in consumer package goods,” Journal of Consumer Research, 21, 408–18.
----, and Russell S. Winer (1995), “Empirical generalization from reference price
research,” Marketing Science, 14(3), G161–69.
Kamakura, Wagner A., Byung-Do Kim, and Jonathan Lee (1996), “Modeling preference
and structural heterogeneity in consumer choice,” Marketing Science, 15(2), 152–72.
----, and Gary J. Russell (1989), “A probabilistic choice model for market segmentation
and elasticity structure,” Journal of Marketing Research, 26, 379–90.
Kannan, P. K., and Gordon P. Wright (1991), “Modeling and testing structured markets:
A nested logit approach,” Marketing Science, 10(1), 58–82.
Klein, Noreen M., and Janet E. Oglethorpe (1987), “Cognitive reference points in
consumer decision making,” in Advances in Consumer Research, Melanie Wallendorf and Paul
Anderson, eds. Provo: UT: Association for Consumer Research.
Krishnamurthi, Lakshman, Tridib Mazumdar, and S. P. Raj (1992), “Asymmetric
response to price in consumer brand choice and purchase quantity decisions,” Journal of
Consumer Research, 19, 387–400.
----, and S. P. Raj (1991), “An empirical analysis of the relationship between brand
loyalty and consumer price elasticity,” Marketing Science, 10(2), 172–83.
Lichtenstein, Donald R., and William O. Bearden (1989), “Contextual influences on
42
perceptions of merchant-supplied reference prices,” Journal of Consumer Research, 16, 55–66.
Linton, Oliver B., and Jens Perch Nielsen (1995), “A kernel method of estimating
structured nonparametric regression models,” Biometrika, 84(2), 469–73.
Mazumdar, Tridib, and Purushottam Papatla (1995), “Loyalty differences in the use of
internal and external reference prices,” Marketing Letters, 6(2), 111–22.
----, and ---- (2000), “An investigation of reference price segments,” Journal of Marketing
Research, 17, 246–58.
Monroe, Kent B. (1971), “Measuring price thresholds by psychophysics and latitudes of
acceptance,” Journal of Marketing Research, 8, 460–64.
Pauwels, Koen H., Franses, Philip Hans and Srinivasan, Shuba (2004), “Reference-based
Transitions in Short-run Price Elasticity,” ERIM Report Series Reference No. ERS-2003-095-
MKT.
Rajendran, K. N., and Gerard J. Tellis (1994), “Contextual and temporal components of
reference price,” Journal of Marketing, 58, 22–34.
Raman, Kalyan, and Frank M. Bass (2002), “A general test of reference price theory in
the presence of threshold effects,” Tijdschrift voor Economie en Management, 47, 205–26.
Sawyer, Alan G., and Peter R. Dickson (1984), “Psychological perspectives on consumer
response to sales promotion,” in Research on Sales Promotion: Collected Papers, Katherine E.
Jocz, ed. Cambridge, MA: Marketing Science Institute.
Sherif, Carolyn. W. (1963), “Social categorization as a function of latitude of acceptance
and series range,” Journal of Abnormal and Social Psychology, 67(2), 148–56.
Sherif, Muzafer, and Carl I. Hovland (1961), Social Judgment. New Haven: Yale
University Press.
Thaler, Richard (1980), “Toward a positive theory of consumer choice,” Journal of
43
44
Economic Behavior and Organization, 1, 39–60.
---- (1985), “Mental accounting and consumer choice,” Marketing Science, 4(3), 199–214.
Tversky, Amos, and Daniel Kahneman (1981), “The framing of decisions and the
psychology of choice,” Science, 211, 453–58.
Urbany, Joel E., William O. Bearden, and Dan C. Weilbaker (1988), “The effect of
plausible and exaggerated reference prices on consumer perceptions and price research,” Journal
of Consumer Research, 15, 95–110.
----, and Peter R. Dickson (1990), “Prospect theory and pricing decision,” Journal of
Behavioral Economics, 19(1), 69–81.
Winer, Russell S. (1985), “A price vector model of demand for consumer durables:
Preliminary developments,” Marketing Science, 4, 74-90.
---- (1986), “A reference price model of brand choice for frequently purchased products,”
Journal of Consumer Research, 13, 250–56.
---- (1988), “Behavioral perspective on pricing: Buyers’ subjective perceptions of price
revisited,” in Issues in Pricing, T. M. Devinney, ed. Lexington, MA: Lexington Books.
Wricke, Martin, Andreas Herrmann, and Franz Huber (2000), “Behavioral pricing,” WiSt,
12, 692–97.
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024 "Modeling the FIBOR/EURIBOR Swap Term Structure: An Empirical Approach" by Oliver Blaskowitz, Helmut Herwartz and Gonzalo de Cadenas Santiago, April 2005.
025 "Duality Theory for Optimal Investments under Model Uncertainty" by Alexander Schied and Ching-Tang Wu, April 2005.
026 "Projection Pursuit For Exploratory Supervised Classification" by Eun-Kyung Lee, Dianne Cook, Sigbert Klinke and Thomas Lumley, May 2005.
027 "Money Demand and Macroeconomic Stability Revisited" by Andreas Schabert and Christian Stoltenberg, May 2005.
028 "A Market Basket Analysis Conducted with a Multivariate Logit Model" by Yasemin Boztuğ and Lutz Hildebrandt, May 2005.
029 "Utility Duality under Additional Information: Conditional Measures versus Filtration Enlargements" by Stefan Ankirchner, May 2005.
030 "The Shannon Information of Filtrations and the Additional Logarithmic Utility of Insiders" by Stefan Ankirchner, Steffen Dereich and Peter Imkeller, May 2005.
031 "Does Temporary Agency Work Provide a Stepping Stone to Regular Employment?" by Michael Kvasnicka, May 2005.
032 "Working Time as an Investment? – The Effects of Unpaid Overtime on Wages, Promotions and Layoffs" by Silke Anger, June 2005.
033 "Notes on an Endogenous Growth Model with two Capital Stocks II: The Stochastic Case" by Dirk Bethmann, June 2005.
034 "Skill Mismatch in Equilibrium Unemployment" by Ronald Bachmann, June 2005.
035 "Uncovered Interest Rate Parity and the Expectations Hypothesis of the Term Structure: Empirical Results for the U.S. and Europe" by Ralf Brüggemann and Helmut Lütkepohl, April 2005.
036 "Getting Used to Risks: Reference Dependence and Risk Inclusion" by Astrid Matthey, May 2005.
037 "New Evidence on the Puzzles. Results from Agnostic Identification on Monetary Policy and Exchange Rates." by Almuth Scholl and Harald Uhlig, July 2005.
038 "Discretisation of Stochastic Control Problems for Continuous Time Dynamics with Delay" by Markus Fischer and Markus Reiss, August 2005.
039 "What are the Effects of Fiscal Policy Shocks?" by Andrew Mountford and Harald Uhlig, July 2005.
040 "Optimal Sticky Prices under Rational Inattention" by Bartosz Maćkowiak and Mirko Wiederholt, July 2005.
041 "Fixed-Prize Tournaments versus First-Price Auctions in Innovation Contests" by Anja Schöttner, August 2005.
042 "Bank finance versus bond finance: what explains the differences between US and Europe?" by Fiorella De Fiore and Harald Uhlig, August 2005.
043 "On Local Times of Ranked Continuous Semimartingales; Application to Portfolio Generating Functions" by Raouf Ghomrasni, June 2005.
044 "A Software Framework for Data Based Analysis" by Markus Krätzig, August 2005.
045 "Labour Market Dynamics in Germany: Hirings, Separations, and Job-to-Job Transitions over the Business Cycle" by Ronald Bachmann, September 2005.
SFB 649, Spandauer Straße 1, D-10178 Berlin http://sfb649.wiwi.hu-berlin.de
This research was supported by the Deutsche
Forschungsgemeinschaft through the SFB 649 "Economic Risk".
046 "Paternal Uncertainty and the Economics of Mating, Marriage, and Parental Investment in Children" by Dirk Bethmann and Michael Kvasnicka, September 2005.
047 "Estimation and Testing for Varying Coeffcients in Additive Models with Marginal Integration " by Lijian Yang, Byeong U. Park, Lan Xue and Wolfgang Härdle, September 2005.
048 "Zeitarbeit in Deutschland: Trends und Perspektiven" by Michael C. Burda and Michael Kvasnicka, September 2005.
049 "Courtesy and Idleness: Gender Differences in Team Work and Team Competition" by Radosveta Ivanova-Stenzel and Dorothea Kübler, September 2005.
050 "Do Factor Shares Reflect Technology?" by Benjamin Bental and Domonique Demougin, September 2005.
051 "Optimal investments for risk- and ambiguity-averse preferences: A duality approach" by Alexander Schied, September 2005.
052 "Relational Contracts and Job Design" by Anja Schöttner, September 2005. 053 "Explicit characterization of the super-replication strategy in financial
markets with partial transaction costs" by Imen Bentahar and Bruno Bouchard, October 2005.
054 "Aid Effectiveness and Limited Enforceable Conditionality" by Almuth Scholl, August 2005.
055 "Limited Enforceable International Loans, International Risk Sharing and Trade" by Almuth Scholl, August 2005.
056 "Stock Markets and Business Cycle Comovement in Germany before World War I: Evidence from Spectral Analysis" by Albrecht Ritschl and Martin Uebele, November 2005.
057 "An empirical test of theories of price valuation using a semiparametric approach, reference prices, and accounting for heterogeneity" by Yasemin Boztuğ and Lutz Hildebrandt, November 2005.
SFB 649, Spandauer Straße 1, D-10178 Berlin http://sfb649.wiwi.hu-berlin.de
This research was supported by the Deutsche
Forschungsgemeinschaft through the SFB 649 "Economic Risk".